the goal of data reduction

The goal of Data Reduction• From a series of diffraction images (films), obtain a file containing the

intensity (I) and standard deviation ((I)) for each reflection, hkl.

H K L I 0 0 4 3295.4 174.00 0 8 482.1 28.70 0 12 9691.0 500.70 0 16 1743.9 67.40 0 20 5856.0 221.00 0 24 14066.5 436.20 0 28 9936.3 311.70 0 36 8409.8 273.40 0 40 790.5 32.80 0 44 103.4 18.4. . . . .. . . . .. . . . .37 7 0 28.5 16.237 7 1 110.1 10.937 7 2 337.4 13.337 7 3 98.5 10.637 7 4 25.9 10.7

Set of filmsFinal intensities

1. Index2. Integrate3.Merge

2,0,0

3,0,-1

4,0,-25,0,-3

5,0,-45,0,-5

4,0,-63,0,-7

2,0,-7

1,0,-80,0,-8

-1,0,-8

Indexing

Assign an h,k,l coordinate to each reflection of the first image.

Integration

Within a spot, sum up the intensity of each pixel.Repeat for each spot on each film.

Merge

Average (merge together) symmetry related reflections.

Plane L=0 K

H

?-H,-K, L=-K, H, L= K,-H, L= H, K,-L= H,-K,-L= K, H,-L=-K,-H,-L=-H,-K,-L= K,-H,-L=-K, H,-L=-H, K, L=-K,-H, L= K, H, L

Plane L=0 K

H

K,-H,L

H,K,L

-H,K,L

-K,-H,L

K,H,L

-H,-K,L

H,-K,L

-K,H,L

Three steps• From a series of diffraction images, obtain a file containing the

intensity (I) and standard deviation ((I)) for each reflection, hkl.

2,0,03,0,-14,0,-25,0,-35,0,-45,0,-5

4,0,-63,0,-72,0,-7

1,0,-80,0,-8-1,0,-8

Set of films

1. index

2. integrate

3. merge

K,-H,L

H,K,L

-H,K,L

-K,-H,L

K,H,L

-H,-K,L

H,-K,L

-K,H,L

Final intensities

H K L I 0 0 4 3295.4 174.00 0 8 482.1 28.70 0 12 9691.0 500.70 0 16 1743.9 67.40 0 20 5856.0 221.00 0 24 14066.5 436.20 0 28 9936.3 311.70 0 36 8409.8 273.40 0 40 790.5 32.80 0 44 103.4 18.4. . . . .. . . . .. . . . .37 7 0 28.5 16.237 7 1 110.1 10.937 7 2 337.4 13.337 7 3 98.5 10.637 7 4 25.9 10.7

2,0,0

3,0,-1

4,0,-25,0,-3

5,0,-45,0,-5

4,0,-63,0,-7

2,0,-7

1,0,-80,0,-8

-1,0,-8

Indexing

How do we find the correct h,k,l coordinate of each reflection?

What’s the h,k,l of this spot?

3 lattice p

oin

ts in

a* d

irection

2 lattice points in

b* direction

For a given spot on the film, we simply have to trace the diffracted ray back to the reciprocal latticepoint (h,k,l)

The answer is HKL=3,2,2

What parameters must be defined to complete this construction?

15 parameters must be determined to index a spot.

The wavelength of the incident radiation

Coordinates (X,Y) of the direct beam

Coordinates (X,Y) for the spot position

Unit cell parametersa,b,c,

The orientation of the unit cell axes with respect to the laboratory axes ().

Which of the 15 parameters are set or known? Which are unknown?

The wavelength of the incident radiation

Coordinates (X,Y) of the direct beam

Coordinates (X,Y) for the spot position

Unit cell parametersa,b,c,

The orientation of the unit cell axes with respect to the laboratory axes ().

How is the unit cell and crystal orientation determined?

Acta Cryst. (1999), D55, 1690-1695

Figure 1.

Choose principle axes by inspection

One dimensional Fourier transforms (7300 orientations)

Find all pairs of spots that can be connected by a vector of given orientation, but any length.

e.g.

One dimensional Fourier transforms (7300 orientations)

Find all pairs of spots that can be connected by a vector of given orientation, but any length.

e.g.

Figure 3. You will find some vector lengths are represented in the diffraction pattern much more frequently than others.These vector lengths differ by integral multiples of one particular value…corresponding to the unit cell dimension.

Figure 3.

One dimensional Fourier transforms (7300 orientations)

Figure 4.

A group of 30 possible non-linear vectors are calculated.

3 vectors at a time are combined to give a basis set of direct-space primitive unit cells.

For each combination of 3 vectors, a distortion index is evaluated which describes how the observed fitted reciprocal lattice deviates from the 14 Bravais lattices.

A chi squared statistics describes the deviations of the observed reflections from the theoretical lattice.

Lattice parameters determined

Xdisplay and Peak Search

1) Display first image in your data set with xdisplay.xdispccd images/my.img

2) Press “Peak Search”. Red circles indicate position of prominent peaks (spots).

3) Evaluate whether you need more or fewer peaks.

4) Pres “OK”

5) Spot positions (x,y) are written to a file “peaks.file.”

Peaks.file• 7777 0.0 0.0 1 1 height X Y frame• 13 2695.7 1350.5 1 1• 27 2669.5 1062.4 1 1• 16 2570.6 1143.5 1 1• 26 2569.4 1302.4 1 1• 30 2562.5 1592.5 1 1• 32 2554.5 1902.4 1 1• 32 2524.5 1103.4 1 1• 22 2514.5 1523.8 1 1• 12 2503.4 1316.6 1 1• 21 2494.5 1949.5 1 1• 15 2492.5 1923.4 1 1• 35 2488.5 1721.5 1 1• 17 2483.5 1870.6 1 1• 12 2479.4 1212.5 1 1• 32 2465.5 1452.5 1 1• 15 2456.4 638.4 1 1• 13 2444.7 900.7 1 1• 14 2437.6 1183.4 1 1• 23 2436.4 1969.4 1 1• Etc…………………………………………..

Run autoindexing script

The autoindexing script is simply titled “a.”

Type “denzo: in the terminal window to start the program Denzo. Then type @a

Select a space group with desired Bravais Lattice (e.g. new space group P4)

Predicted pattern should match observed diffraction pattern.

“go” to refine

Necessary to index film 2 from scratch?

film3? etc?

1o

Film 1, exposed over 1 to 2 degrees Film 2, exposed over 2 to 3 degrees

The first film provides all the parameters need to predict the location of every spot on every film.

The wavelength of the incident radiation

Coordinates (X,Y) of the direct beam

Coordinates (X,Y) for the spot position

Unit cell parametersa,b,c,

The orientation of the unit cell axes with respect to the laboratory axes ().

Integration

Adjust integration box size and background box size.spot elliptical 0.6 0.6background elliptical 0.7 0.7

Paste parameters into integration script (integ.dat).

Insert refined unit cell and crystal orientation parameters into

integration script (integ.dat).Type “list” to obtain refined paramers..

Integrated intensities are written to .x files

Film 1, exposed over 1 to 2 degrees

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1……………………………………………………….

One .x file for each film

Film 1, exposed over 1 to 2 degrees

Film 2, exposed over 2 to 3 degrees

Film 180, exposed over 180 to 181 degrees

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 52.3 50.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix-

29 -20 33 1 212.3 220.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

prok_001.img

prok_001.x

prok_002.img prok_180.img

prok_002.x prok_180.x

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

360 frames, 1 degree

rotation each

With .x files, we can map intensities onto a reciprocal

lattice

1) Accuracy will improve if we Merge multiple observations of the same reciprocal lattice point

2) But, we must test if rotational symmetry exists between lattice points.

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

h k l flag I(profit) I(prosum) 2 (I) cos incid. X pix Y pix

29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.629 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061

prok_001 -> 360.x

Is it Laue group 422 Or Laue group 4?

P4 H, K,L-H,-K,L-K, H,L K,-H,L

H, K,-L H,-K,-L K, H,-L-K,-H,-L

P422

Test existenceof 4-fold

symmetry

Test existence of4-fold Symmetry

andPerpendicular

2-fold symmetry

j observations of the reflection 30 22 6

<I>= (100 + 500 + 300) / 3 = 300

j H K L I

1 30 22 6 100

2 -30 -22 6 500

3 22 -30 6 300

Rsym= |Ij-<I>|

Ij

Discrepancy between symmetry related reflections

|Ij-<I>| = |100-300|+|500-300|+|300-300|

= 200 + 200 + 0 = 400Ij = 100 + 500 + 300

= 900

Rsym = 400/900 = 0.44 = 44%

Discrepancy between symmetry related reflections (Rsym) increases with increasing resolution. Why?

Shell Rsym

100-5.0Å 0.04

5.0-3.0Å 0.06

3.0-2.0Å 0.08

2.0-1.7Å 0.15

Statistics are analyzed as a function of resolution (N shells).

Average I/ decreases with increasing resolution

High resolution shells with I/ <2 should be discarded.

Shell I/

100-5.0 Å 20.0

5.0-3.0 Å 10.0

3.0-2.0 Å 7.0

2.0-1.7 Å 3.0

SIGNAL TO NOISE RATIO (I/)

COMPLETENESS?

What percentage of reciprocalLattice was measured for a givenResolution limit?

Better than 90% I hope.

Shell completeness

100-5.0Å 99.9%

5.0-3.0Å 95.5%

3.0-2.0Å 89.0%

2.0-1.7Å 85.3%

Overall 92.5%

Assignment

Effect of mosaicity and wavelength spread

Figure 5.

Figure 6.